In an equilateral triangle, when you the altitude it forms 2 right angled triangles.
lets consider one of them.
let the altitude be x (which is to be found out)
hypotenuse is a and the base is (a/2) (since the altitude bisects the base of the equilateral triangle)
by applying pythogorus theore, x= sqrt[a^2- (a/2)^2]